
We again apply \(\mathbf {vec}(\mathbf {A}\mathbf {B}\mathbf {C}) = (\mathbf {C}^\top \otimes \mathbf {A})\mathbf {vec}(\mathbf {B})\)  [1]} to obtain
\(p_{SA}^* &= \max _{\mathbf {V}_i} -\sum _{i=1}^n \mathcal {L}^*\left(\mathbf {V}_i, \mathbf {Y}_i\right) \nonumber \\\mathrm {s.t.}~& \max _{\begin{array}{c}\Vert \mathbf {w}_1\Vert _2\le 1\end{array}} \left\Vert  \sum _{i=1}^N ({\mathbf {X}}_i \otimes \mathbf {V}_i^\top ) ((\mathbf {X}_i\otimes {\mathbf {X}}_i) \mathbf {w}_1)_+\right\Vert _2.\) 
